The distribution of spacings between the fractional parts of

We study the distribution of spacings between the fractional parts of $n^d\alpha$. For $\alpha$ of high enough Diophantine type we prove a necessary and sufficient condition for $n^d\alpha \mathrm{m}\mathrm{o}\mathrm{d}1,1\le n\le N,$ to be Poissonian as $N\to \mathrm{\infty }$ along a suitable subsequence.